Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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55 views

Do cycle graphs determine groups up to isomorphism?

This erroneous version of the wikipedia article for Cycle Graphs stated that the cycle graph of different groups could be the same. The immediate error is that it stated that the cycle graph of ...
2
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1answer
54 views

Confusion with Centers, Conjugacy Classes, and Normal Subgroups

Ok this is a bit of a soft question, but I am in my first semester of algebra and getting continually confused by (what seems to me) some competing sets. Let $G$ be a group The center of $G$ is ...
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21 views

Relations between the Latin squares of order n and the groups of order n.

Given a group $g$ of order $n\in \mathbb{N}$, it is clear that its Cayley table has the Latin square property. That is, every row and column of the Cayley table contains precisely the elements of $g$. ...
4
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1answer
61 views

Is this parabolic induction?

In one of my previous questions, @PL. explained the idea of parabolic induction. In my bachelor thesis I used a technique quite like this, to find all irreducible representations of a certain group. I ...
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2answers
48 views

Showing Sylow $p$-groups are Abelian if $n_p=2p+1$

An old qual question on Sylow $p$-groups: Assume that $p$ is an odd prime and $G$ is a finite simple group with exactly $2p+1$ Sylow $p$-groups. Prove that the Sylow $p$-groups of $G$ are abelian. ...
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21 views

What is the purpose of the almost maximal and $ p $-supersoluble subgroup?

Suppose that $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose that there is an element $ y \in H $ such that $ H = \langle y \rangle L $ for any almost maximal subgroup L of $ H $; then $ G $ ...
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1answer
46 views

How do we identify $\mathfrak{R}$-automorphisms of a group?

If $G$ is a finite group, a bijection $f\colon G\to G$ is called a (normed) $\boldsymbol{\mathfrak{R}}$-automorphism if $f$ maps subgroups of $G$ to subgroups of $G$, and $f(gH) = f(g) f(H)$ for any ...
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2answers
69 views

Set of generators for $A_n$, the alternating group.

The problem is this: Prove that $A_n = \langle (123),(124),\ldots,(12n)\rangle$. I had cogitated this problem for quite awhile, and haven't been able to come up with anything. The only good idea ...
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3answers
51 views

Which inverse multiplicative groups modulo $n$ are cyclic or not

I've found nothing about this in my book neither in the internet. Also the wikipedia article about inverse multiplicative modulo $n$ is poor. So, I need prove that $$\mathbb Z_n^*$$ is cyclic for ...
4
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3answers
100 views

Representation of an abelian group

Without using the structure theorem, how do I prove b? I struggle with the proof of injectivity. Any tips? Problem: Let $G$ be a finite Abelian group. (a) Prove that the group homomorphisms $\chi ...
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2answers
50 views

Exercise about finding group isomorphisms

So, i've just learned about groups homomorphisms and isomorphisms. I know that an homormorphism betweet two groups is a function such that $$\phi(a\square b) = \phi(a)\star \phi(b)$$ And when the ...
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1answer
60 views

All primes $p$ for which $x^{2} +1$ irreducible over $\mathbb{F}_{p}$

Let $\mathbb{F}_{p}$ be the field with $p$ elements. Determine all primes $p$ for which $x^{2}+1$ is irreducible over $\mathbb{F}_{p}$. Here's what I've got. I think this might be finished but I'm ...
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2answers
52 views

Subgroup of a group with 24 elements.

Suppose that $G$ is a finite group of order $24$, which has four $3$-sylow subgroups. We know that may contain $1$ or $3$ 2-sylow subgroup. How can I prove that there only exists one $2$-sylow ...
3
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2answers
124 views

Consequence of First Homomorphism Theorem?

Let $\phi:G\to\bar G$ be a surjective homomorphism with kernel $N$. Then the first homomorphism theorem tells us that $G/N\cong\bar G$. My question is this: Lagrange's theorem also tells us that ...
3
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1answer
107 views

A question on subgroups of a finite group

Let $G$ be a finite group. Let $K$ and $H$ be subgroups of $G$, where $K$ is normal in $G$ and $\gcd([G:H],|K|)=1$. Prove that $K$ is a subgroup of $H$. So far we found that $o(K)$ divides ...
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1answer
20 views

Questions about terminology (transpositions)

A cycle with only two elements is called a transposition. For example, the permutation of $\{1, 2, 3, 4\}$ that sends $1$ to $1$, $2$ to $4$, $3$ to $3$ and $4$ to $2$ is a transposition ...
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2answers
38 views

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help ...
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2answers
33 views

Show that left cosets partition the group

I know how to prove that it happens, by proving that the left coset definition actually is an equivalence relation. Then, it's proved that it partitions the set, since equivalence relations do it. ...
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1answer
38 views

Finite (cardinality) modules over a PID

Let $R$ be a principal ideal domain with $|R|=\infty$. Suppose $M$ is an $R$-module such that $2 \le |M| < \infty$. What properties does this imply about $R$? Background: I was hoping that $R\cong ...
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2answers
82 views

Prove that $H$ is a normal subgroup of $G$

This is a problem from the book "Berkeley Problems in Mathematics": Let $G$ be a group of order $120$, let $H$ be a subgroup of order $24$, and assume that there is at least one coset of $H$ ...
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0answers
28 views

If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots \unlhd G_{s} = G $ is a chief series of $ G $.

The $ p $-Fitting Subgroup of $ G $ is the maximal normal $ p $-nilpotent subgroup Of $ G $ and write it $ F_{p}(G) $. If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots ...
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1answer
19 views

How to generate the icosahedral groups $I$ and $I_h$?

The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their ...
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1answer
59 views

How can I show that the characters in sense of irreducible representations are the same as the character maps from the burnside matrices?

My Task is: Let G be a finite group. 1. Let $C_1 = \{e\}, C_2,..., C_k$ be the conjugacy classes, and let $v_1,..., v_k$ be the normalised eigenvectors of the Burnside matrices of G, then for all s ...
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1answer
53 views

Let $G$ be a group of finite order, $H$ and $K$ subgroups so that $H \unlhd G$; $K \unlhd HK \unlhd G$ and $(|H|,|K|) = 1$. Show that $K \unlhd G$.

I've been trying to solve this for a little while. I know that $H\cap K = \{1\}$ because of their orders and from the isomorphism theorems I know that $ HK /K \simeq H$. I've been trying to see if ...
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1answer
48 views

Suppose $ G = G_{1}G_{2} $. For any prime $ p $, there exist $ P , P_{1} , P_{2} $ such that $ P = P_{1}P_{2} $

Suppose $ G = G_{1}G_{2} $. For any prime $ p $, there exist $ P , P_{1} , P_{2} $ such that $ P = P_{1}P_{2} $, where $ P \in Syl_{p}(G) $ and $ P_{i} \in Syl_{p}(G_{i}) $ , $ i = 1,2 $. The proof ...
2
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4answers
51 views

Let $H, K$ be two subgroups of $G$. If $|H| = 12$ and $|K|=17$ then $H \cap K = \{e\}$

My reasoning: Since $|K| = 17$ and $17$ is prime, then any subgroup of $K$ is cyclic. Also, the order of any subgroup must divide the order of the group. But since the subgroups of $K$ must have an ...
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3answers
84 views

Proving that $gHg^{-1}$ is a subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$ with $\operatorname{ord}(H)=k$ I need to prove that $gHg^{-1}$ is a subgroup of $G$ $\color{grey}{(gHg^{-1}=\{ghg^{-1}\mid g\in G, h\in H\})}$ My ...
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155 views

$G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p

Let $G$ be a non abelian group of order $p^3$, with $p$ prime. I'm proving that $Z(G)$ (its center) is of order $p$. I already know how to do it by saying that its order can't be $p^3$, nor 1, and if ...
3
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1answer
28 views

Finding subgroups of $\mathbb{Z}_{20}$

I need to find all the subgroups of $\mathbb{Z}_{20}$ My attempt: $\mathbb{Z}_{20}$ is cyclic $\Longrightarrow$ all the subgroups will be also cyclic, according to Lagrangh the order of the ...
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0answers
40 views

If $ Q = \langle y \rangle X $ for some element $ y $, then $ \vert N \vert = p $ if $ p $ is odd and $ \vert N \vert \leq 4 $ if $ p = 2 $.

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $ which has a complement $ X $ in $ Q $. If $ Q = \langle y \rangle X $ for some element $ y $, then ...
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4answers
46 views

Calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$

I'm trying to calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$. I'm really bad with groups so I'd appreciate a verification of my conclusion: If $m$ is even then $\forall x\in \mathbb Z_m$ we get ...
3
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1answer
57 views

Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$

I need to prove or disprove: Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$ My attempt: No, there isn't isomorphism, because if it did then $S_4$ would have an element of order $8$, ...
0
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1answer
58 views

Number of elements of order $11$ in group of order $1331$

Let $G$ be a group of order $1331$. Prove that $G$ has at least $11$ elements of order $11$. $|G|=1331=11^3$ So by First Sylow's theorem, there exists a Sylow $11$-subgroup of G. By Third Sylow's ...
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2answers
60 views

Prove or disprove $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$

I need to prove or disprove: To the group $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$ My attempt: I just wrote all the details that I know: element in $A_5$ should be in form like ...
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4answers
716 views

Product of all elements in finite group

Question: If $G$ is a finite group such that the product of its elements (each chosen only once) is always $1$, independent of the ordering in the product, what can we say about $G$? I was trying to ...
2
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0answers
44 views

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $ which has a complement $ X $ in $ Q $. If $ Q = \langle y \rangle X $ for some element $ y $, then ...
3
votes
2answers
87 views

Finding order of $gag^{-1}$ in $G$ if $a^2=e\in G$

Let $G$ be a group, the order of $G$ is even, let $a \in G$, $a^2=e$ I need to find the order of $gag^{-1}$ in $G$ My attempt: ...
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1answer
43 views

Cardinality of automorphism groups of groups of order $p^4$.

As far as I know there is no classification of the automorphism groups of groups of order $p^4$. (see ...
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2answers
95 views

are these two finite groups with different presentation isomorphic?

Consider two groups $$\langle x,y \, | \, x^4=y^5=1 ,yxy=x \rangle$$ and $$ \langle a,b \, | \, a^{10}=1,b^2=a^5,aba=b \rangle.$$ I think they are isomorphic, but I can't show it, it will be great if ...
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3answers
394 views

Noncyclic Abelian Group of order 51

The problem is to prove or disprove that there is a noncyclic abelian group of order $51$. I don't think such a group exists. Here is a brief outline of my proof: Assume for a contradiction that ...
2
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2answers
74 views

Computing Factor Group

I am reading John Fraleigh's First Course in Abstract Algebra, $\S$36 on the Second Isomorphism Theorem which says that if $H < G$ and $N \triangleleft G$, then $$(HN)/N \cong H/(H \cap N).$$ He ...
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1answer
37 views

Find the order of $\tau^{100}$

Let $\tau= \left( \begin{array}{ccc} ...
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2answers
59 views

Why normal subgroup chains in Galois theory

I have began understanding Galois theory and I had a question regarding the relationships of normal subgroups to field extensions. So given an irreducible polynomial over the rationals $$a_1 + a_2x ...
3
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3answers
84 views

Show that $G/H\cong\mathbb{R}^*$

Let $G:= \bigg\{\left( \begin{array}{ccc} a & b \\ 0 & a \\ \end{array} \right)\mid a,b \in \mathbb{R},a\ne 0\bigg\}$ Let $H:= \bigg\{\left( \begin{array}{ccc} 1 & b \\ 0 & 1 \\ ...
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2answers
41 views

Finding subgroups of $\mathbb{Z}_{13}^*$

I need to find all nontrivial subgroups of $G:=\mathbb{Z}_{13}^*$ (with multiplication without zero) My attempt: $G$ is cyclic so the order of subgroup of $G$ must be $2,3,4,6$ Now to look for ...
4
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2answers
64 views

Is a non-trivial finite perfect group of order 4n?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial. Question: Is a non-trivial finite perfect group of ...
1
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1answer
42 views

$G$ finite, $P < G$ a Sylow p-subgroup, $N_{G}(P)$ the normalizer is contained in $H < G$, show $N_{G}(H)=H$

Let $G$ be a finite group and $P<G$ be a Sylow p-subgroup. Let $N_{G}(P)$ be the normalizer of $P$ in $G$. Let $H<G$ be a subgroup containing $N_{G}(P)$. Prove that $N_{G}(H)=H$. I've been ...
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2answers
70 views

Name and role of a particular finite group?

The group generated by the functions $x\mapsto 1/x$ and $x\mapsto 1-x$ with composition of functions as the group operation is a non-abelian group with only six elements (listed below). Does this ...
2
votes
1answer
26 views

Group theory exercise from “Rational Points on Elliptic Curves”

Let $A$ be an abelian group and for every $m \geq 1$, let $A_m$ be the set of elements $P$ of $A$ such that $mP=0$. Now, suppose that $A$ has order $M^2$ and that for every integer $m$ dividing ...
3
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2answers
66 views

Subgroups of $\mathbb{Z}_{13}^*$ using “GAP language”

How can I know the nontrivial subgroups of $\mathbb{Z}_{13}^*$ (with multiplication without zero) using GAP language